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In 2022, Öztürk defined the nearness equivalence classes and nearness cosets of the nearness groups (see [10]). Furthermore, he showed a nearness group G induced by a nearness subgroup H can not be written two different decompositions of G by separate left (right) cosets [10]. In other words, if H is a nearness subgroup of a nearness group G, then G may be shown as a union of different non-discrete left (right) cosets of H in G. Also, it is given that Lagrange's theorem does not work in the nearness subgroups as usual subgroups. Our purpose in this paper is to introduce normal nearness subgroups and quotient nearness groups. Besides which, it is given an example of a nearness group that has nearness subgroup but not normal nearness subgroup. Also, a criterion that satisfies the condition of being a normal subgroup is investigated. It is shown that the multiplication of the two right near-cosets (left near-cosets) is again a right near-coset (left near-coset) is valid in normal nearness subgroups under certain conditions. Moreover, examples of normal nearness subgroups and quotient nearness groups are given. [ABSTRACT FROM AUTHOR]

The notions of hybrid subsemigroups and hybrid left (resp., right) ideals in semigroups are introduced, and several properties are investigated. Using these notions, characterizations of subsemigroups and left (resp., right) ideals are discussed. The concept of hybrid product is also introduced, and characterizations of hybrid subsemigroups and hybrid left (resp., right) ideals are considered by using the notion of hybrid product. Relations between hybrid intersection and hybrid product are displayed. [ABSTRACT FROM AUTHOR]